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<div class="subTitle">org.apache.commons.math3.analysis.polynomials</div>
<h2 title="Class PolynomialsUtils" class="title">Class PolynomialsUtils</h2>
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<pre>public class <span class="strong">PolynomialsUtils</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">A collection of static methods that operate on or return polynomials.</div>
<dl><dt><span class="strong">Since:</span></dt>
  <dd>2.0</dd></dl>
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<h3>Method Summary</h3>
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<caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
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<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
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<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createChebyshevPolynomial(int)">createChebyshevPolynomial</a></strong>(int&nbsp;degree)</code>
<div class="block">Create a Chebyshev polynomial of the first kind.</div>
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<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createHermitePolynomial(int)">createHermitePolynomial</a></strong>(int&nbsp;degree)</code>
<div class="block">Create a Hermite polynomial.</div>
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<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createJacobiPolynomial(int,%20int,%20int)">createJacobiPolynomial</a></strong>(int&nbsp;degree,
                      int&nbsp;v,
                      int&nbsp;w)</code>
<div class="block">Create a Jacobi polynomial.</div>
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<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createLaguerrePolynomial(int)">createLaguerrePolynomial</a></strong>(int&nbsp;degree)</code>
<div class="block">Create a Laguerre polynomial.</div>
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<td class="colFirst"><code>static <a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#createLegendrePolynomial(int)">createLegendrePolynomial</a></strong>(int&nbsp;degree)</code>
<div class="block">Create a Legendre polynomial.</div>
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<td class="colFirst"><code>static double[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialsUtils.html#shift(double[],%20double)">shift</a></strong>(double[]&nbsp;coefficients,
     double&nbsp;shift)</code>
<div class="block">Compute the coefficients of the polynomial \(P_s(x)\)
 whose values at point <code>x</code> will be the same as the those from the
 original polynomial \(P(x)\) when computed at <code>x + shift</code>.</div>
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<h3>Methods inherited from class&nbsp;java.lang.<a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3>
<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li>
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<h3>Method Detail</h3>
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<h4>createChebyshevPolynomial</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a>&nbsp;createChebyshevPolynomial(int&nbsp;degree)</pre>
<div class="block">Create a Chebyshev polynomial of the first kind.
 <p><a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev
 polynomials of the first kind</a> are orthogonal polynomials.
 They can be defined by the following recurrence relations:</p><p>
 \(
    T_0(x) = 1 \\
    T_1(x) = x \\
    T_{k+1}(x) = 2x T_k(x) - T_{k-1}(x)
 \)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd>
<dt><span class="strong">Returns:</span></dt><dd>Chebyshev polynomial of specified degree</dd></dl>
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<h4>createHermitePolynomial</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a>&nbsp;createHermitePolynomial(int&nbsp;degree)</pre>
<div class="block">Create a Hermite polynomial.
 <p><a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite
 polynomials</a> are orthogonal polynomials.
 They can be defined by the following recurrence relations:</p><p>
 \(
  H_0(x) = 1 \\
  H_1(x) = 2x \\
  H_{k+1}(x) = 2x H_k(X) - 2k H_{k-1}(x)
 \)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd>
<dt><span class="strong">Returns:</span></dt><dd>Hermite polynomial of specified degree</dd></dl>
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<h4>createLaguerrePolynomial</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a>&nbsp;createLaguerrePolynomial(int&nbsp;degree)</pre>
<div class="block">Create a Laguerre polynomial.
 <p><a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre
 polynomials</a> are orthogonal polynomials.
 They can be defined by the following recurrence relations:</p><p>
 \(
   L_0(x) = 1 \\
   L_1(x) = 1 - x \\
   (k+1) L_{k+1}(x) = (2k + 1 - x) L_k(x) - k L_{k-1}(x)
 \)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd>
<dt><span class="strong">Returns:</span></dt><dd>Laguerre polynomial of specified degree</dd></dl>
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<h4>createLegendrePolynomial</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a>&nbsp;createLegendrePolynomial(int&nbsp;degree)</pre>
<div class="block">Create a Legendre polynomial.
 <p><a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre
 polynomials</a> are orthogonal polynomials.
 They can be defined by the following recurrence relations:</p><p>
 \(
   P_0(x) = 1 \\
   P_1(x) = x \\
   (k+1) P_{k+1}(x) = (2k+1) x P_k(x) - k P_{k-1}(x)
 \)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd>
<dt><span class="strong">Returns:</span></dt><dd>Legendre polynomial of specified degree</dd></dl>
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<h4>createJacobiPolynomial</h4>
<pre>public static&nbsp;<a href="../../../../../../org/apache/commons/math3/analysis/polynomials/PolynomialFunction.html" title="class in org.apache.commons.math3.analysis.polynomials">PolynomialFunction</a>&nbsp;createJacobiPolynomial(int&nbsp;degree,
                                        int&nbsp;v,
                                        int&nbsp;w)</pre>
<div class="block">Create a Jacobi polynomial.
 <p><a href="http://mathworld.wolfram.com/JacobiPolynomial.html">Jacobi
 polynomials</a> are orthogonal polynomials.
 They can be defined by the following recurrence relations:</p><p>
 \(
    P_0^{vw}(x) = 1 \\
    P_{-1}^{vw}(x) = 0 \\
    2k(k + v + w)(2k + v + w - 2) P_k^{vw}(x) = \\
    (2k + v + w - 1)[(2k + v + w)(2k + v + w - 2) x + v^2 - w^2] P_{k-1}^{vw}(x) \\
  - 2(k + v - 1)(k + w - 1)(2k + v + w) P_{k-2}^{vw}(x)
 \)
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>degree</code> - degree of the polynomial</dd><dd><code>v</code> - first exponent</dd><dd><code>w</code> - second exponent</dd>
<dt><span class="strong">Returns:</span></dt><dd>Jacobi polynomial of specified degree</dd></dl>
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<h4>shift</h4>
<pre>public static&nbsp;double[]&nbsp;shift(double[]&nbsp;coefficients,
             double&nbsp;shift)</pre>
<div class="block">Compute the coefficients of the polynomial \(P_s(x)\)
 whose values at point <code>x</code> will be the same as the those from the
 original polynomial \(P(x)\) when computed at <code>x + shift</code>.
 <p>
 More precisely, let \(\Delta = \) <code>shift</code> and let
 \(P_s(x) = P(x + \Delta)\).  The returned array
 consists of the coefficients of \(P_s\).  So if \(a_0, ..., a_{n-1}\)
 are the coefficients of \(P\), then the returned array
 \(b_0, ..., b_{n-1}\) satisfies the identity
 \(\sum_{i=0}^{n-1} b_i x^i = \sum_{i=0}^{n-1} a_i (x + \Delta)^i\) for all \(x\).</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>coefficients</code> - Coefficients of the original polynomial.</dd><dd><code>shift</code> - Shift value.</dd>
<dt><span class="strong">Returns:</span></dt><dd>the coefficients \(b_i\) of the shifted
 polynomial.</dd></dl>
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